/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 54 Multiply. $$ (2 x-y)(2 x+y) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 54

Multiply. $$ (2 x-y)(2 x+y) $$

Short Answer

Expert verified
The result is \(4x^2 - y^2\).

Step by step solution

01

Identify the Pattern

First, recognize that the expression \((2x-y)(2x+y)\) is in the form \((a-b)(a+b)\), which represents the difference of squares pattern \((a^2 - b^2)\).
02

Apply the Difference of Squares Formula

Using the formula \((a-b)(a+b) = a^2 - b^2\), substitute \(a = 2x\) and \(b = y\). This gives: \((2x)^2 - (y)^2\).
03

Calculate and Simplify

Now calculate \((2x)^2\) and \(y^2\). \((2x)^2 = 4x^2\) and \(y^2\) remains \(y^2\). Therefore, the expression becomes \(4x^2 - y^2\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares Pattern
In algebra, the difference of squares is a special algebraic pattern used to simplify expressions quickly. This pattern is based on the formula \((a-b)(a+b) = a^2 - b^2\). It works because the middle terms in the expanded form cancel each other out. For example, when we multiply \((2x-y)\) and \((2x+y)\), we notice it fits this pattern:
  • If \(a = 2x\), then \(a^2\) is \((2x)^2 = 4x^2\).
  • If \(b = y\), then \(b^2\) is \(y^2\).
When these are plugged into the formula, the expression simplifies to \(4x^2 - y^2\). Understanding this pattern helps with quick computations.
Polynomial Multiplication
Multiplying polynomials is a key skill in algebra, often using the distributive property. Essentially, each term in one polynomial multiplies by each term in the other. For instance, with polynomials \((2x-y)\) and \((2x+y)\), you distribute as follows:
  • \(2x \times 2x = 4x^2\)
  • \(2x \times y = 2xy\)
  • \(-y \times 2x = -2xy\)
  • \(-y \times y = -y^2\)
The middle terms, \(2xy\) and \(-2xy\), cancel each other, leaving you with \(4x^2 - y^2\). This shows how polynomial multiplication ties into the alternate solution using the difference of squares.
Factoring Expressions
Factoring is the process of breaking down expressions into products of simpler expressions. This technique is invaluable when simplifying equations or solving polynomial expressions.To factor a difference of squares, you reverse the difference of squares formula:
  • If you start with \(4x^2 - y^2\), you identify it as \(a^2 - b^2\), implying \((a-b)(a+b)\).
  • You determine \(a = 2x\) and \(b = y\), thus factoring to \((2x-y)(2x+y)\).
Factoring is a powerful tool for simplifying and solving polynomial equations by making them more manageable.
Understanding Mathematical Expressions
Mathematical expressions are combinations of variables, numbers, and operators that represent a value. Understanding them is crucial in algebra, especially as they can be manipulated in various ways to solve problems.Expressions like \((2x-y)(2x+y)\) can be simplified or expanded depending on what you're trying to achieve:
  • To simplify: Identify patterns such as difference of squares to make calculations faster.
  • To expand: Use polynomial multiplication to understand the structure of the expression.
Deep knowledge of mathematical expressions allows you to navigate algebraic problems more efficiently, breaking them down into simpler components.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A piece of quarter-round molding is \((13 x-7)\) inches long. If a piece \((2 x+2)\) inches long is removed, express the length of the remaining piece of molding as a polynomial in \(x\).

Evaluate each expression using exponential rules. Write each result in standard form. $$ \left(4 \times 10^{-10}\right)\left(7 \times 10^{-9}\right) $$

Subtract. $$ 3 x-(5 x-9) $$

Simplify each expression. Assume that variables represent positive integers. $$ \left(3 y^{2 z}\right)^{3} $$

Match each expression on the left with its simplification on the right. Not all letters on the right must be used and a letter may be used more than once. a. \(3 y\) b. \(9 y-6 y^{2}\) c. \(10 x\) d. \(25 x^{2}\) e. \(10 x-6\) f. none of these $$ (15 x-3)-(5 x-3) $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.